# Properties

 Label 3234.2.a.bd Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.8236200137$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - 2 * q^5 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9} - 2 q^{10} - q^{11} + q^{12} + (\beta - 4) q^{13} - 2 q^{15} + q^{16} - 2 q^{17} + q^{18} + ( - 3 \beta - 2) q^{19} - 2 q^{20} - q^{22} + (2 \beta - 2) q^{23} + q^{24} - q^{25} + (\beta - 4) q^{26} + q^{27} + ( - 4 \beta + 4) q^{29} - 2 q^{30} + (3 \beta - 2) q^{31} + q^{32} - q^{33} - 2 q^{34} + q^{36} + ( - 2 \beta - 2) q^{37} + ( - 3 \beta - 2) q^{38} + (\beta - 4) q^{39} - 2 q^{40} + (4 \beta - 6) q^{41} + (2 \beta - 2) q^{43} - q^{44} - 2 q^{45} + (2 \beta - 2) q^{46} + ( - 7 \beta - 2) q^{47} + q^{48} - q^{50} - 2 q^{51} + (\beta - 4) q^{52} + (6 \beta - 2) q^{53} + q^{54} + 2 q^{55} + ( - 3 \beta - 2) q^{57} + ( - 4 \beta + 4) q^{58} + (2 \beta - 4) q^{59} - 2 q^{60} + ( - \beta - 4) q^{61} + (3 \beta - 2) q^{62} + q^{64} + ( - 2 \beta + 8) q^{65} - q^{66} + ( - 2 \beta - 4) q^{67} - 2 q^{68} + (2 \beta - 2) q^{69} - 4 \beta q^{71} + q^{72} + (4 \beta - 6) q^{73} + ( - 2 \beta - 2) q^{74} - q^{75} + ( - 3 \beta - 2) q^{76} + (\beta - 4) q^{78} + (6 \beta - 8) q^{79} - 2 q^{80} + q^{81} + (4 \beta - 6) q^{82} + (5 \beta + 2) q^{83} + 4 q^{85} + (2 \beta - 2) q^{86} + ( - 4 \beta + 4) q^{87} - q^{88} + ( - 3 \beta - 8) q^{89} - 2 q^{90} + (2 \beta - 2) q^{92} + (3 \beta - 2) q^{93} + ( - 7 \beta - 2) q^{94} + (6 \beta + 4) q^{95} + q^{96} - 7 \beta q^{97} - q^{99} +O(q^{100})$$ q + q^2 + q^3 + q^4 - 2 * q^5 + q^6 + q^8 + q^9 - 2 * q^10 - q^11 + q^12 + (b - 4) * q^13 - 2 * q^15 + q^16 - 2 * q^17 + q^18 + (-3*b - 2) * q^19 - 2 * q^20 - q^22 + (2*b - 2) * q^23 + q^24 - q^25 + (b - 4) * q^26 + q^27 + (-4*b + 4) * q^29 - 2 * q^30 + (3*b - 2) * q^31 + q^32 - q^33 - 2 * q^34 + q^36 + (-2*b - 2) * q^37 + (-3*b - 2) * q^38 + (b - 4) * q^39 - 2 * q^40 + (4*b - 6) * q^41 + (2*b - 2) * q^43 - q^44 - 2 * q^45 + (2*b - 2) * q^46 + (-7*b - 2) * q^47 + q^48 - q^50 - 2 * q^51 + (b - 4) * q^52 + (6*b - 2) * q^53 + q^54 + 2 * q^55 + (-3*b - 2) * q^57 + (-4*b + 4) * q^58 + (2*b - 4) * q^59 - 2 * q^60 + (-b - 4) * q^61 + (3*b - 2) * q^62 + q^64 + (-2*b + 8) * q^65 - q^66 + (-2*b - 4) * q^67 - 2 * q^68 + (2*b - 2) * q^69 - 4*b * q^71 + q^72 + (4*b - 6) * q^73 + (-2*b - 2) * q^74 - q^75 + (-3*b - 2) * q^76 + (b - 4) * q^78 + (6*b - 8) * q^79 - 2 * q^80 + q^81 + (4*b - 6) * q^82 + (5*b + 2) * q^83 + 4 * q^85 + (2*b - 2) * q^86 + (-4*b + 4) * q^87 - q^88 + (-3*b - 8) * q^89 - 2 * q^90 + (2*b - 2) * q^92 + (3*b - 2) * q^93 + (-7*b - 2) * q^94 + (6*b + 4) * q^95 + q^96 - 7*b * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{11} + 2 q^{12} - 8 q^{13} - 4 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{19} - 4 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{24} - 2 q^{25} - 8 q^{26} + 2 q^{27} + 8 q^{29} - 4 q^{30} - 4 q^{31} + 2 q^{32} - 2 q^{33} - 4 q^{34} + 2 q^{36} - 4 q^{37} - 4 q^{38} - 8 q^{39} - 4 q^{40} - 12 q^{41} - 4 q^{43} - 2 q^{44} - 4 q^{45} - 4 q^{46} - 4 q^{47} + 2 q^{48} - 2 q^{50} - 4 q^{51} - 8 q^{52} - 4 q^{53} + 2 q^{54} + 4 q^{55} - 4 q^{57} + 8 q^{58} - 8 q^{59} - 4 q^{60} - 8 q^{61} - 4 q^{62} + 2 q^{64} + 16 q^{65} - 2 q^{66} - 8 q^{67} - 4 q^{68} - 4 q^{69} + 2 q^{72} - 12 q^{73} - 4 q^{74} - 2 q^{75} - 4 q^{76} - 8 q^{78} - 16 q^{79} - 4 q^{80} + 2 q^{81} - 12 q^{82} + 4 q^{83} + 8 q^{85} - 4 q^{86} + 8 q^{87} - 2 q^{88} - 16 q^{89} - 4 q^{90} - 4 q^{92} - 4 q^{93} - 4 q^{94} + 8 q^{95} + 2 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^10 - 2 * q^11 + 2 * q^12 - 8 * q^13 - 4 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 - 4 * q^19 - 4 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^24 - 2 * q^25 - 8 * q^26 + 2 * q^27 + 8 * q^29 - 4 * q^30 - 4 * q^31 + 2 * q^32 - 2 * q^33 - 4 * q^34 + 2 * q^36 - 4 * q^37 - 4 * q^38 - 8 * q^39 - 4 * q^40 - 12 * q^41 - 4 * q^43 - 2 * q^44 - 4 * q^45 - 4 * q^46 - 4 * q^47 + 2 * q^48 - 2 * q^50 - 4 * q^51 - 8 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 4 * q^57 + 8 * q^58 - 8 * q^59 - 4 * q^60 - 8 * q^61 - 4 * q^62 + 2 * q^64 + 16 * q^65 - 2 * q^66 - 8 * q^67 - 4 * q^68 - 4 * q^69 + 2 * q^72 - 12 * q^73 - 4 * q^74 - 2 * q^75 - 4 * q^76 - 8 * q^78 - 16 * q^79 - 4 * q^80 + 2 * q^81 - 12 * q^82 + 4 * q^83 + 8 * q^85 - 4 * q^86 + 8 * q^87 - 2 * q^88 - 16 * q^89 - 4 * q^90 - 4 * q^92 - 4 * q^93 - 4 * q^94 + 8 * q^95 + 2 * q^96 - 2 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
1.00000 1.00000 1.00000 −2.00000 1.00000 0 1.00000 1.00000 −2.00000
1.2 1.00000 1.00000 1.00000 −2.00000 1.00000 0 1.00000 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3234.2.a.bd yes 2
3.b odd 2 1 9702.2.a.cy 2
7.b odd 2 1 3234.2.a.bc 2
21.c even 2 1 9702.2.a.ci 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bc 2 7.b odd 2 1
3234.2.a.bd yes 2 1.a even 1 1 trivial
9702.2.a.ci 2 21.c even 2 1
9702.2.a.cy 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{13}^{2} + 8T_{13} + 14$$ T13^2 + 8*T13 + 14 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 8T + 14$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 4T - 14$$
$23$ $$T^{2} + 4T - 4$$
$29$ $$T^{2} - 8T - 16$$
$31$ $$T^{2} + 4T - 14$$
$37$ $$T^{2} + 4T - 4$$
$41$ $$T^{2} + 12T + 4$$
$43$ $$T^{2} + 4T - 4$$
$47$ $$T^{2} + 4T - 94$$
$53$ $$T^{2} + 4T - 68$$
$59$ $$T^{2} + 8T + 8$$
$61$ $$T^{2} + 8T + 14$$
$67$ $$T^{2} + 8T + 8$$
$71$ $$T^{2} - 32$$
$73$ $$T^{2} + 12T + 4$$
$79$ $$T^{2} + 16T - 8$$
$83$ $$T^{2} - 4T - 46$$
$89$ $$T^{2} + 16T + 46$$
$97$ $$T^{2} - 98$$